Geometric Properties of Families of Galois Representations

Abstract

This thesis concerns families of Galois representations arising as etale local systems on a variety over a number field or a p-adic field. The first part of the thesis studies families of Galois representations of number fields in the context of the relative Fontaine-Mazur conjecture. The conjecture predicts that a p-adic etale local system that satisfies the de Rham condition arises from algebraic geometry and in particular such a local system is part of a compatible system of l-adic etale local systems with various primes l. We show the existence of a compatible system in some cases. We also discuss the finiteness of the field generated over Q by Frobenius traces of a local system at closed points. The second part of the thesis studies families of Galois representations of p-adic fields in the context of the relative p-adic Hodge theory. Sen attached to each p-adic Galois representation of a p-adic field a multiset of numbers called generalized Hodge-Tate weights. We consider a p-adic local system on a rigid analytic variety over a p-adic field and show that the multiset of generalized Hodge-Tate weights of the local system is constant. We also discuss basic properties of Hodge-Tate sheaves on a rigid analytic variety.